9.2 Linear Equations in Two Variables

A linear equation in two variables relates two first-power unknowns, such as 2x + 3y = 12 or y = -(2/3)x + 4. College Board's Algebra description for this testing point includes creating the equation from a constraint, connecting tables, graphs, and algebra, interpreting a solution or a coefficient in context, and writing a line from two points or from a point and a slope.

One equation is not a unique pair (x, y) unless a second independent equation is given. That second-equation story is 10.1 Systems of Two Linear Equations. Here you name the line, the slope, the intercepts, and any one pair that sits on the line. OpenExamPrep's independent teaching for this skill uses original numbers; it is not an official College Board worksheet.

Three forms you should recognize

These are the same line written differently. Fluency is converting, not memorizing three unrelated recipes. Parallel lines share slope and never meet; that fact becomes central in systems. If a stem says a line through (0, 1) is parallel to y = 3x + 5, copy the slope 3 and write y = 3x + 1. Perpendicular slopes (negative reciprocals) can appear in later algebra courses; keep PSAT 8/9 work on rate, intercepts, and equivalent forms unless a question clearly gives that extra geometric hint.

Worked example: standard form to slope-intercept and intercepts

Start with 2x + 3y = 12.

Solve for y:

3y = -2x + 12

y = -(2/3)x + 4

So m = -2/3 and b = 4. The graph crosses the y-axis at (0, 4). Each time x increases by 3, y decreases by 2.

Intercepts from standard form (set the other variable to 0):

• y = 0: 2x = 12, so x = 6. The x-intercept is (6, 0).
• x = 0: 3y = 12, so y = 4. The y-intercept is (0, 4).

Two points are enough to sketch the line: (6, 0) and (0, 4).

Check a third point. If x = 3, then y = -(2/3)(3) + 4 = -2 + 4 = 2. And 2(3) + 3(2) = 6 + 6 = 12. The pair (3, 2) is a solution. So are (0, 4) and (6, 0). Infinitely many solutions sit on that one line.

Worked example: 5x + 2y = 20

• x-intercept: y = 0 gives 5x = 20, so x = 4. Point (4, 0).
• y-intercept: x = 0 gives 2y = 20, so y = 10. Point (0, 10).
• Slope-intercept: 2y = -5x + 20, so y = -(5/2)x + 10.

Slope -5/2 means down 5, right 2. Check (4, 0): -(5/2)(4) + 10 = -10 + 10 = 0.

Slope as a rate; intercept as a start

In y = mx + b, m is the constant rate: how much y changes when x increases by 1. b is the starting value when x = 0.

Bike rental. A shop charges $8 to walk in, then $3 per hour. Let h be hours and C be cost in dollars.

C = 3h + 8

• Slope 3: each extra hour adds $3.
• Intercept 8: at 0 hours the desk fee is already $8.

Four hours: C = 3(4) + 8 = 12 + 8 = 20 dollars. A $29 bill: 3h + 8 = 29, so 3h = 21, h = 7. Check: 21 + 8 = 29.

Plumber. Visit fee $45 plus $70 per hour: c = 70h + 45. A $255 bill: 70h + 45 = 255, so 70h = 210, h = 3. Check: 70 · 3 + 45 = 210 + 45 = 255.

Mixing slope and intercept is the usual trap. A bowling graph that starts at $12 when x = 0 (shoe rental) and rises to $16 when x = 1 (one game) has intercept 12 and slope 4. The slope is dollars per game. The intercept is not the price of a game.

Create an equation from a table

From 0 to 1, cost rises 7; from 1 to 2, rises 7; from 2 to 3, rises 7. The rate is constant, so the relationship is linear. When h = 0, C = 15, so the intercept is 15.

C = 7h + 15

Check the last row: 7(3) + 15 = 21 + 15 = 36.

If a row broke the pattern—say 0 maps to 15, 1 maps to 22, and 2 maps to 31—you would not force C = 7h + 15. Constant rate is the test for this chapter. Nonlinear tables belong with later Advanced Math, not with a forced line.

Second table:

Change in x is 2; change in y is 4; so m = 4/2 = 2. The intercept is 5. y = 2x + 5 Check: 2(4) + 5 = 13.

Two points, then point-slope

A line passes through (1, 8) and (5, 0).

m = (0 - 8)/(5 - 1) = -8/4 = -2

Point-slope with (1, 8):

y - 8 = -2(x - 1)

y - 8 = -2x + 2

y = -2x + 10

Check the second point: -2(5) + 10 = 0.

Another pair: (2, 7) and (6, 15).

m = (15 - 7)/(6 - 2) = 8/4 = 2

y - 7 = 2(x - 2)

y = 2x - 4 + 7

y = 2x + 3

Check: 2(6) + 3 = 15, and 2(2) + 3 = 7.

Point-slope when you already have m

Through (3, -2) with slope 4:

y - (-2) = 4(x - 3)

y + 2 = 4x - 12

y = 4x - 14

Check: 4(3) - 14 = 12 - 14 = -2.

A two-variable constraint in a story

Pretzel bags cost $2 and drinks cost $3. A family spends $18:

2p + 3d = 18

If they buy 3 pretzel bags: 2(3) + 3d = 18, so 6 + 3d = 18, 3d = 12, d = 4. Check: 6 + 12 = 18.

The equation 2p + 3d = 18 by itself still has many other solutions, such as (0, 6) and (9, 0). The extra sentence they buy 3 pretzel bags is what pins down d. Interpreting coefficients: the 2 is dollars per pretzel bag; the 3 is dollars per drink; 18 is the total spent. That reading—constant, variable, or term in context—is a stated Algebra skill on the SAT Suite, including PSAT 8/9.

Graph connections and Desmos

The line y = -6x - 32 has y-intercept (0, -32), because substituting x = 0 leaves y = -32. Slope -6 means a drop of 6 in y for each increase of 1 in x. If a question asks only for the intercept value y, an SPR answer is -32 (include the minus sign).

Desmos check: graph the equation you wrote and click the intercepts. If the story said an $8 start and $3 per hour, y = 3x + 8 should hit (0, 8) and (4, 20). If it does not, the algebra is wrong—not the rental desk. Use the graph as a check after setup, the same habit as in 9.1 Linear Equations in One Variable.

/practice/psat-89Practice questions with detailed explanations